BOUNDARY-ELEMENT METHOD FOR THE CALCULATION OF ELECTRONIC STATES IN SEMICONDUCTOR NANOSTRUCTURES

We hate developed a boundary-element method to treat the single-partic le electronic properties of semiconductor nanostructures that consist of piecewise homogeneous materials of arbitrary shapes. Green's-functi on techniques are used to derive integral equations that determine the se electronic properties. These equations involve integrals over the b oundaries between the homogeneous regions, and they are discretized an d solved numerically. In effect, this approach changes a partial diffe rential equation with boundary conditions in d independent variables i nto an integral equation in d-1 independent variables, which leads to its efficiency. For bound states these methods are used to calculate e igenenergies, for scattering states to calculate differential cross se ctions, and for both bound and scattering states to calculate spectral density functions and wave functions. For such systems, are show that this method generally provides improved calculational efficiency as c ompared to alternative approaches such as plane-wave expansions, finit e-difference methods, or finite-element methods and that it is more ef fective in treating highly excited states than are these methods. Illu strative examples are given here for several systems whose potentials are functions of two variables, such cis quantum wires or patterned tw o-dimensional electron gases.